How large dimension guarantees a given angle?
In mathematics it is a typical task to prove the existence of certain
given patterns in large enough sets. In this talk we will investigate
such a problem in geometric measure theory.
Let S be a subset of a Euclidean space. Given an angle β, we would like to prove theorems claiming that S must contain provided that the Hausdorff dimension of S is large enough. (We say that S contains the angle β if there exist three distinct points in S such that the enclosed angle is β.)
On the other hand, we construct large dimensional sets not containing given angles.
Some of the results are joint work with Keleti, Kiss, Maga, Máthé, Mattila and Strenner.